Moment constraint:  
Some controls cannot be directly defined by off-the-shelf models, and the moment constraint [1]}, [2]} is one of them. Given a mapping \(\mathbf {\gamma }: \mathcal {X} \rightarrow \mathbb {R}^{K}\)  (e.g., \(\mathbf {\gamma }\)  can be a classifier that outputs the probability simplex), the moment constraint defines the target distribution \(p(\mathbf {x}|\mathcal {C})\)  as
\(p(\mathbf {x}|\mathcal {C}) = \underbrace{\mathop {\arg \min }_{p(\mathbf {x}|\mathcal {C})} \mathbb {D}_{\mathsf {KL}}(p(\mathbf {x}|\mathcal {C})\Vert p_{\mathbf {x}}(\mathbf {x}))}_{\text{Deviation from the pre-trained distribution}}, \quad \text{s.t. } \underbrace{\mathbb {E}_{\mathbf {x}\sim p(\mathbf {x}|\mathcal {C})}\big [\mathbf {\gamma }(\mathbf {x})\big ] = \mathbf {\mu }}_{\text{Moment constraint}},\) 
